Summary: We perform consistently the Gupta-Bleuler-Dirac quantization for a two-dimensional boson with parameter \((\alpha)\) on the circle, the boundary of the circular droplet. For \(\alpha=1\), we obtain the chiral (holomorphic) constraints. Using the representation of Bargmann-Fock space and the Schrödinger picture, we construct the holomorphic wave function. In order to interpret this function, we construct the coherent state representation by using the infinite-dimensional translation \((W_\infty)\) symmetry for each Fourier (edge) mode. The \(\alpha=1\) chiral wave function explains the neutral edge states for the integer quantum Hall effect very well. In the case of \(\alpha=-1\), we obtain a new wave function which may describe the higher modes (radial excitations) of edge states. The charged edge states are described by the \(|\alpha|\neq 1\) wave function. Finally, the application of our model to the fractional quantum Hall effect is discussed.